Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-9x^3 - 72x^2 + 81x}{x^3 + 17x^2 + 72x}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-9x(x^2 + 8x - 9)} {x(x^2 + 17x + 72)} $ $ p = -\dfrac{9x}{x} \cdot \dfrac{x^2 + 8x - 9}{x^2 + 17x + 72} $ Simplify: $ p = - 9 \cdot \dfrac{x^2 + 8x - 9}{x^2 + 17x + 72}$ Since we are dividing by $x$ , we must remember that $x \neq 0$ Next factor the numerator and denominator. $ p = - 9 \cdot \dfrac{(x + 9)(x - 1)}{(x + 9)(x + 8)}$ Assuming $x \neq -9$ , we can cancel the $x + 9$ $ p = - 9 \cdot \dfrac{x - 1}{x + 8}$ Therefore: $ p = \dfrac{ -9(x - 1)}{ x + 8 }$, $x \neq -9$, $x \neq 0$